# Prove this block matrices are similar

Prove that the block matrices
$\left( \begin{array}{cc} AB & 0\\ B & 0\\ \end{array} \right)$
and
$\left( \begin{array}{cc} 0 & 0\\ B & BA\\ \end{array} \right)$
are similar.

Where $\mathbf{K}$ is any field, $A\in \mathbf{K}^{m\times n}$, $B\in \mathbf{K}^{n\times m}$
and both matrices in $\mathbf{K}^{(m+n)\times (m+n)}$.

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#### Solutions Collecting From Web of "Prove this block matrices are similar"

From Fuzhen Zhang, “Matrix Theory – Basic Results and Techniques” (Springer, 1999), page 52:

\begin{align*}
\begin{bmatrix} {\rm I}_m & A \\ 0 & {\rm I}_n \end{bmatrix} \begin{bmatrix} 0 & 0 \\ B & 0 \end{bmatrix} = \begin{bmatrix} AB & 0 \\ B & 0 \end{bmatrix}, \\
\begin{bmatrix} 0 & 0 \\ B & 0 \end{bmatrix} \begin{bmatrix} {\rm I}_m & A \\ 0 & {\rm I}_n \end{bmatrix} = \begin{bmatrix} 0 & 0 \\ B & BA \end{bmatrix}, \\
\end{align*}
follows that
$$\begin{bmatrix} {\rm I}_m & A \\ 0 & {\rm I}_n \end{bmatrix}^{-1} \begin{bmatrix} AB & 0 \\ B & 0 \end{bmatrix} \begin{bmatrix} {\rm I}_m & A \\ 0 & {\rm I}_n \end{bmatrix} = \begin{bmatrix} 0 & 0 \\ B & BA \end{bmatrix}.$$